Is this a paradox in Special Relativity?

[Shackleford]

So, it's critical to make the observation that the observer in the Fixed Frame never leaves his respective inertial frame, but the observer in the Moving frame does in fact change inertial frames. Because of those non-inertial frames/phases, it's valid to put the relativistic effect/corrections on the Moving frame and look at it from the "fixed" Fixed frame, i.e. the Moving frame is the moving clock which runs slowly and yields the shorter time interval. You then also have to make corrections for length contraction.

I don't see the point of using the word "journey" like this. I would call that curve the object's "world line", and use words like "journey" only when I'm talking about the corresponding sequence of events in the real world.

Is that not what a journey is? That's how I'm looking at it - a series of infinitesimally-small events that constitute the curve. Any point on the world line or curve is an event at a specific point in space and time. There's just a bunch of them depending on how you sub-divide the time interval along that curve.

 

[Fredrik]

So, it's critical to make the observation that the observer in the Fixed Frame never leaves his respective inertial frame, but the observer in the Moving frame does in fact change inertial frames.

I'm not a fan of phrases like "leaves his inertial frame". I recommend that you never say that an object is "in" an inertial frame or that it "changes" inertial frames. Instead, say that it's stationary in an inertial frame and that it changes velocity in that same inertial frame. If it changes velocity, it now has a different comoving inertial frame.

Inertial frames are just global coordinate systems. A global coordinate system is a function from spacetime into $$\mathbb R^4$$. So the inertial frames are just functions that assign coordinates to events. You can't be "in" one, but you have a velocity in all of them.

Is that not what a journey is? That's how I'm looking at it - a series of infinitesimally-small events that constitute the curve.

I'm just saying that if you travel to South Africa, that would be a journey (unless you happen to live there). I don't see a reason to use the word "journey" for the mathematical representation of that journey when the we already have a standard term for it: "world line".

 

[jtbell]

Did you mean 1.2 ms?

Ugh, you're right. I don't remember how I could have possibly come up with 0.17 ms this morning. I should have waited until after I had my quota of coffee. I've corrected it and the other related numbers. Thanks for catching that.

 

[GRDixon]

Start: Two clocks sitting next to each other on Earth are synchronized.
Next: One clock is put into a rocket and launched on a high-speed trip to Alpha Centuari and back.
Finish: The two clocks are sitting next to each other again on earth.

Which is ahead and which is behind at the finish?

The rocket clock is behind (less time has elapsed on it) upon its return. On the way out to Alpha Centauri the rocket occupant can measure the rate of the Earth clock (using the clocks distributed in his own rest frame) and will deduce that the Earth clock runs more slowly than those at rest in his own rest frame. But when he changes rest frames upon reaching Alpha Centauri, he will admit that the clocks in his initial rest frame are actually not synchronized with one another, etc., and that it is actually his clock that has been running slowly during the trip. The same situation repeats on his way back to Earth. By way of a real example, short half-life mesons last much longer (don't decay) when shot around the loop of an accelerator.